BOUNDARY BEHAVIOUR OF QUOTIENTS OF MARTIN KERNELS

Given two intersecting domains, we investigate the boundary behaviour of the quotient of Martin kernels of each domain. To this end, we give a characterization of minimal thinness for a difference of two subdomains in terms of Martin kernels of each domain. As a consequence of our main theorem, we obtain the boundary growth of the Martin kernel of a Lipschitz domain, which corresponds to earlier results for the boundary decay of the Green function for a Lipschitz domain investigated by Burdzy, Carroll and Gardiner.

On minimal thinness, reduced functions and Green potentials

Let Ω be an open connected subset of the unit disc U, let E = U\Ω and let {Ωk} be a Whitney decomposition of U. If z(Q) is the centre of the “square” Q, if T is the unit circle and t = dist.(Q, T), we considerwhere Ek = E ∩ Qk and c(Ek) is the capacity of Ek. We prove that the set E is minimally thin at τ ∈ T in U if and only if W(τ)< ∞. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjögren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsall [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.